Preheat your oven to 400°F (200°C) and line a rimmed baking sheet with parchment paper.
Pat the chicken drumsticks thoroughly dry with paper towels to ensure the skin gets crispy.
In a bowl, toss the drumsticks with olive oil, sea salt, and black pepper until evenly coated.
Place the chicken on the prepared baking sheet and roast for 25 to 30 minutes, turning once halfway through, until the skin is golden and the internal temperature reaches 165°F.
While the chicken roasts, mince the garlic and grate the ginger.
In a small saucepan over low heat, whisk together the honey, tamari, garlic, and ginger until the sauce slightly thickens and becomes aromatic.
Place the broccoli florets in a steamer basket over boiling water and steam for 5 minutes until tender-crisp and bright green.
Remove the chicken from the oven and brush each drumstick generously with the honey-garlic glaze.
Serve the glazed chicken drumsticks immediately alongside the steamed broccoli.